Problem: What is the extraneous solution to these equations? $\dfrac{x^2 + 69}{x - 9} = \dfrac{150}{x - 9}$
Solution: Multiply both sides by $x - 9$ $ \dfrac{x^2 + 69}{x - 9} (x - 9) = \dfrac{150}{x - 9} (x - 9)$ $ x^2 + 69 = 150$ Subtract $150$ from both sides: $ x^2 + 69 - (150) = 150 - (150)$ $ x^2 + 69 - 150 = 0$ $ x^2 - 81 = 0$ Factor the expression: $ (x + 9)(x - 9) = 0$ Therefore $x = -9$ or $x = 9$ At $x = 9$ , the denominator of the original expression is 0. Since the expression is undefined at $x = 9$, it is an extraneous solution.